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<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.ellint.ellint_d"></a><a class="link" href="ellint_d.html" title="Elliptic Integral D - Legendre Form">Elliptic Integral D - Legendre
      Form</a>
</h3></div></div></div>
<h5>
<a name="math_toolkit.ellint.ellint_d.h0"></a>
        <span class="phrase"><a name="math_toolkit.ellint.ellint_d.synopsis"></a></span><a class="link" href="ellint_d.html#math_toolkit.ellint.ellint_d.synopsis">Synopsis</a>
      </h5>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">ellint_d</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_d</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_d</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_d</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_d</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>

<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<h5>
<a name="math_toolkit.ellint.ellint_d.h1"></a>
        <span class="phrase"><a name="math_toolkit.ellint.ellint_d.description"></a></span><a class="link" href="ellint_d.html#math_toolkit.ellint.ellint_d.description">Description</a>
      </h5>
<p>
        These two functions evaluate the incomplete elliptic integral <span class="emphasis"><em>D(φ,
        k)</em></span> and its complete counterpart <span class="emphasis"><em>D(k) = D(π/2, k)</em></span>.
      </p>
<p>
        The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
        type calculation rules</em></span></a> when the arguments are of different
        types: when they are the same type then the result is the same type as the
        arguments.
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_d</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
        Returns the incomplete elliptic integral:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint_d.svg"></span>

        </p></blockquote></div>
<p>
        Requires <span class="emphasis"><em>k<sup>2</sup>sin<sup>2</sup>(phi) &lt; 1</em></span>, otherwise returns the result
        of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        (outside this range the result would be complex).
      </p>
<p>
        The final <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
        be used to control the behaviour of the function: how it handles errors,
        what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">policy
        documentation for more details</a>.
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_d</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_d</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
        Returns the complete elliptic integral <span class="emphasis"><em>D(k) = D(π/2, k)</em></span>
      </p>
<p>
        Requires <span class="emphasis"><em>-1 &lt;= k &lt;= 1</em></span> otherwise returns the result
        of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        (outside this range the result would be complex).
      </p>
<p>
        The final <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
        be used to control the behaviour of the function: how it handles errors,
        what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">policy
        documentation for more details</a>.
      </p>
<h5>
<a name="math_toolkit.ellint.ellint_d.h2"></a>
        <span class="phrase"><a name="math_toolkit.ellint.ellint_d.accuracy"></a></span><a class="link" href="ellint_d.html#math_toolkit.ellint.ellint_d.accuracy">Accuracy</a>
      </h5>
<p>
        These functions are trivially computed in terms of other elliptic integrals
        and generally have very low error rates (a few epsilon) unless parameter
        φ
is very large, in which case the usual trigonometric function argument-reduction
        issues apply.
      </p>
<div class="table">
<a name="math_toolkit.ellint.ellint_d.table_ellint_d_complete_"></a><p class="title"><b>Table 8.66. Error rates for ellint_d (complete)</b></p>
<div class="table-contents"><table class="table" summary="Error rates for ellint_d (complete)">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              </th>
<th>
                <p>
                  GNU C++ version 7.1.0<br> linux<br> double
                </p>
              </th>
<th>
                <p>
                  GNU C++ version 7.1.0<br> linux<br> long double
                </p>
              </th>
<th>
                <p>
                  Sun compiler version 0x5150<br> Sun Solaris<br> long double
                </p>
              </th>
<th>
                <p>
                  Microsoft Visual C++ version 14.1<br> Win32<br> double
                </p>
              </th>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  Elliptic Integral E: Mathworld Data
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0.637ε (Mean = 0.368ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.27ε (Mean = 0.735ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.27ε (Mean = 0.735ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0.637ε (Mean = 0.368ε)</span>
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Elliptic Integral D: Random Data
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0ε (Mean = 0ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.27ε (Mean = 0.334ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.27ε (Mean = 0.334ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.27ε (Mean = 0.355ε)</span>
                </p>
              </td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.ellint.ellint_d.table_ellint_d"></a><p class="title"><b>Table 8.67. Error rates for ellint_d</b></p>
<div class="table-contents"><table class="table" summary="Error rates for ellint_d">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              </th>
<th>
                <p>
                  GNU C++ version 7.1.0<br> linux<br> double
                </p>
              </th>
<th>
                <p>
                  GNU C++ version 7.1.0<br> linux<br> long double
                </p>
              </th>
<th>
                <p>
                  Sun compiler version 0x5150<br> Sun Solaris<br> long double
                </p>
              </th>
<th>
                <p>
                  Microsoft Visual C++ version 14.1<br> Win32<br> double
                </p>
              </th>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  Elliptic Integral E: Mathworld Data
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
                  2.1:</em></span> Max = 0.862ε (Mean = 0.568ε))
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.3ε (Mean = 0.813ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.3ε (Mean = 0.813ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0.862ε (Mean = 0.457ε)</span>
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Elliptic Integral D: Random Data
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
                  2.1:</em></span> Max = 3.01ε (Mean = 0.928ε))
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 2.51ε (Mean = 0.883ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 2.51ε (Mean = 0.883ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 2.87ε (Mean = 0.805ε)</span>
                </p>
              </td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><p>
        The following error plot are based on an exhaustive search of the functions
        domain, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code>
        precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span>
        <span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>.
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../graphs/elliptic_integral_d__double.svg" align="middle"></span>

        </p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../graphs/elliptic_integral_d__80_bit_long_double.svg" align="middle"></span>

        </p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../graphs/elliptic_integral_d____float128.svg" align="middle"></span>

        </p></blockquote></div>
<h5>
<a name="math_toolkit.ellint.ellint_d.h3"></a>
        <span class="phrase"><a name="math_toolkit.ellint.ellint_d.testing"></a></span><a class="link" href="ellint_d.html#math_toolkit.ellint.ellint_d.testing">Testing</a>
      </h5>
<p>
        The tests use a mixture of spot test values calculated using values calculated
        at <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a>, and random
        test data generated using MPFR at 1000-bit precision and a deliberately naive
        implementation in terms of the Legendre integrals.
      </p>
<h5>
<a name="math_toolkit.ellint.ellint_d.h4"></a>
        <span class="phrase"><a name="math_toolkit.ellint.ellint_d.implementation"></a></span><a class="link" href="ellint_d.html#math_toolkit.ellint.ellint_d.implementation">Implementation</a>
      </h5>
<p>
        The implementation for D(φ, k) first performs argument reduction using the
        relations:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="serif_italic"><span class="emphasis"><em>D(-φ, k) = -D(φ, k)</em></span></span>
        </p></blockquote></div>
<p>
        and
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="serif_italic"><span class="emphasis"><em>D(nπ+φ, k) = 2nD(k) + D(φ, k)</em></span></span>
        </p></blockquote></div>
<p>
        to move φ to the range [0, π/2].
      </p>
<p>
        The functions are then implemented in terms of Carlson's integral R<sub>D</sub>
using
        the relation:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/ellint_d.svg"></span>

        </p></blockquote></div>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
</div>
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